Effective diffusion models

Suppose that catalyst pellets in the shape of right-circular cylinders have a measured effectiveness factor of r] when used in a packed-bed reactor for a first-order reaction. In an effort to increase catalyst activity, it is proposed to use a pellet with a central hole of radius i /, < Rp. Determine the best value for RhjRp based on an effective diffusivity model similar to Equation (10.33). Assume isothermal operation ignore any diffusion limitations in the central hole, and assume that the ends of the cylinder are sealed to diffusion. You may assume that k, Rp, and eff are known. 

Work Problem 10.14 using the pore diffusion model rather than the effective diffusivity model. 

The majority of recently published papers are based on this kind of effective diffusion model [7, 10]. In the models of Springer et al. [7] and later Nguyen et al. [10], the local membrane state is determined by the local activity of water which is in thermodynamic equilibrium with surrounding water vapor. Diffusion of water driven by the activity gradient balances the electroosmotic flow. Under stationary conditions this results in a characteristic profile of w across the membrane, with lowest values in the proximity of the anode. 

Following the idea of Yu. Volfkovich, a model of stationary water flows in the membrane with account of porous structure-related aspects and inhomogeneous water distribution was developed [16,83]. This model will be presented in some detail below. Its implications on water-content profiles and current-voltage performance under fuel cell operation conditions will be compared to the effective diffusion models. 

SOLUTION OF MULTICOMPONENT DIFFUSION PROBLEMS USING AN EFFECTIVE DIFFUSIVITY MODEL... 

Example 6.3.1 Computation ofthe Fluxes with an Effective Diffusivity Model... 

Figure 6.1 shows the concentration time history in the diffusion cell for the experiment of Duncan and Toor that was described in detail in Example 5.4.1. The mole fraction of hydrogen predicted by the effective diffusivity model compares well with the experimental data of Duncan (1960). However, the effective diffusivity model suggests that the mole fraction of nitrogen should remain almost constant at approximately 0.5. This is in stark contrast to the experimental data (Fig. 6.1). The results obtained with the effective diffusivity method for nitrogen are completely different from those obtained with the linearized theory. Additional comparisons between the data of Duncan and Toor and the predictions of both the linearized equations and the effective diffusivity models are shown in the triangular diagram in Figure 6.2.

Additional data of Arnold and Toor are compared to the predictions of the linearized equations and of the effective diffusivity models in the triangular diagram in Figure 6.4. Clearly, the agreement with the data is very bad indeed. Thus, we have our second demonstration of the inability of the effective diffusivity method to model systems that exhibit strong diffusional interactions. ... 

Here we use an effective diffusivity model for the diffusion fluxes... 

Figure 8.11. Composition profiles in a Stefan tube. Lines are computed from effective diffusivity model. Data of Carty and Schrodt (1975).

Diffusional interaction effects are quite important in this example. We leave it as an exercise for our readers to determine the molar rates of condensation using an effective diffusivity model. It is worth pointing out, however, that the rates are quite different from those calculated here. 

We see from these figures that the mass transfer models that take diffusional interactions into account are quite a lot better than the effective diffusivity model, which underpredicts the rate of condensation of 2-propanol in every case. However, the effective diffusivity methods give good predictions of the overall temperature drops (Fig. 15.19) although there is little to distinguish any of the models here on this basis. 

Run 7 of the simulations of Modine s experiments is particularly sensitive to the mass transfer model used. This experiment was used as the basis for the flux calculation in Example 10.4.1. For this experiment none of the models does well at predicting the total amount of acetone transfer. The Krishna-Standart method, the linearized theory and the effective diffusivity model predict the wrong direction of mass transfer while the experimental data show that there is net vaporization of acetone, the models predict net condensation. This erroneous prediction comes about in part because of the extremely small magnitude of the acetone flux relative to the total flux. 

Effective diffusivity models should not be used in the determination of the rates of mass transfer in the vapor phase. These models are not justified on theoretical grounds, nor on experimental grounds and their use offers no reduction in the cost of obtaining a solution or any increase in the ease by which that solution is obtained. 

Repeat Example 8.3.1 (equimolar diffusion in a ternary system) using an effective diffusivity method for determining the fluxes and composition profiles. Compare the fluxes calculated with the effective diffusivity model to those obtained in Example... 

In order to calculate the diffusive flux, a suitable mass transfer model must be assumed. T vo categories of models exist (1) interactive models (due to Krishna and Standart [206] and Toor [207]), and (2) noninteractive models, known also as effective diffusivity models. For the interactive models, the diffusion flux j b is... 

Effective diffusion models have also been used to account for intermediate degrees of mixing in the axial direction—see Pavlica and Olson [74] for a useful comprehensive survey. An example of such a model is developed here for the case of a... 

Figure 12.6 Effectiveness factor plot for spherical catalyst pellets based on the effective diffusivity model for a first-order reaction.

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